# Retarded potential

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{{Use American English|date = March 2019}}
{{Short description|Type of potential in electrodynamics}}
{{electromagnetism}}
In [electrodynamics](/source/electrodynamics), the '''retarded potentials''' are the [electromagnetic potential](/source/electromagnetic_potential)s for the [electromagnetic field](/source/electromagnetic_field) generated by [time-varying](/source/Time-variant_system) [electric current](/source/electric_current) or [charge distribution](/source/charge_distribution)s in the past. The fields propagate at the [speed of light](/source/speed_of_light) ''c'', so the delay of the fields connecting [cause and effect](/source/Causality_(physics)) at earlier and later times is an important factor: the signal takes a finite time to propagate from a point in the charge or current distribution (the point of cause) to another point in space (where the effect is measured), see figure below.<ref>{{cite encyclopedia |title=Potentials|last=Rohrlich|first=F|encyclopedia=McGraw Hill Encyclopaedia of Physics|date=1993 |location=New York|editor-last=Parker |editor-first=S.P.|isbn=0-07-051400-3 |page=1072 |edition=2nd|url=https://archive.org/details/mcgrawhillencycl1993park/page/1072/}}</ref> 

==In the Lorenz gauge==
{{main article|Maxwell's equations|Mathematical descriptions of the electromagnetic field}}250px|right|thumb|Position vectors '''r''' and '''r&prime;''' used in the calculation

The starting point is [Maxwell's equations in the potential formulation](/source/Maths_of_EM_field) using the [Lorenz gauge](/source/Lorenz_gauge):

:<math> \Box \varphi = \dfrac{\rho}{\epsilon_0} \,,\quad \Box \mathbf{A} = \mu_0\mathbf{J}</math>

where φ('''r''', ''t'') is the [electric potential](/source/electric_potential) and '''A'''('''r''', ''t'') is the [magnetic vector potential](/source/magnetic_vector_potential), for an arbitrary source of [charge density](/source/charge_density) ρ('''r''', ''t'') and [current density](/source/current_density) '''J'''('''r''', ''t''), and <math>\Box</math> is the [D'Alembert operator](/source/D'Alembert_operator).<ref>Garg, A., ''Classical Electromagnetism in a Nutshell'', 2012, p. 129</ref> Solving these gives the retarded potentials below (all in [SI units](/source/SI_units)).

===For time-dependent fields===

For time-dependent fields, the retarded potentials are:<ref>Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, {{ISBN|978-0-471-92712-9}}</ref><ref>Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, {{ISBN|81-7758-293-3}}</ref>

:<math> \mathrm\varphi (\mathbf r , t) = \frac{1}{4\pi\epsilon_0}\int \frac{\rho (\mathbf r' ,  t_r)}{|\mathbf r - \mathbf r'|}\, \mathrm{d}^3\mathbf r'</math>

:<math>\mathbf A (\mathbf r , t) = \frac{\mu_0}{4\pi}\int \frac{\mathbf J (\mathbf r' ,  t_r)}{|\mathbf r - \mathbf r'|}\, \mathrm{d}^3\mathbf r'\,.</math>

where '''r''' is a [point](/source/position_vector) in space, ''t'' is time,

:<math>t_r = t-\frac{|\mathbf r - \mathbf r'|}{c}</math>

is the [retarded time](/source/retarded_time), and d<sup>3</sup>'''r'''' is the [integration measure](/source/List_of_integration_and_measure_theory_topics) using '''r''''.

From φ('''r''', t) and '''A'''('''r''', ''t''), the fields '''E'''('''r''', ''t'') and '''B'''('''r''', ''t'') can be calculated using the definitions of the potentials:

:<math>-\mathbf{E} = \nabla\varphi +\frac{\partial\mathbf{A}}{\partial t}\,,\quad \mathbf{B}=\nabla\times\mathbf A\,.</math>

and this leads to [Jefimenko's equations](/source/Jefimenko's_equations) and to [Panofsky–Phillips equations](/source/Panofsky%E2%80%93Phillips_equations). The corresponding advanced potentials have an identical form, except the advanced time

:<math>t_a = t+\frac{|\mathbf r - \mathbf r'|}{c}</math>

replaces the retarded time.

===In comparison with static potentials for time-independent fields===

In the case the fields are time-independent ([electrostatic](/source/electrostatic) and [magnetostatic](/source/magnetostatic) fields), the time derivatives in the <math>\Box</math> operators of the fields are zero, and Maxwell's equations reduce to

:<math> \nabla^2 \varphi =-\dfrac{\rho}{\epsilon_0}\,,\quad \nabla^2 \mathbf{A} =- \mu_0 \mathbf{J}\,,</math>

where ∇<sup>2</sup> is the [Laplacian](/source/Laplacian), which take the form of [Poisson's equation](/source/Poisson's_equation) in four components (one for φ and three for '''A'''), and the solutions are:

:<math> \mathrm\varphi (\mathbf{r}) = \frac{1}{4\pi\epsilon_0}\int \frac{\rho (\mathbf r' )}{|\mathbf r - \mathbf r'|}\, \mathrm{d}^3\mathbf r'</math>

:<math>\mathbf A (\mathbf{r}) = \frac{\mu_0}{4\pi}\int \frac{\mathbf J (\mathbf r' )}{|\mathbf r - \mathbf r'|}\, \mathrm{d}^3\mathbf r'\,.</math>

These also follow directly from the retarded potentials.

==In the Coulomb gauge==

In the [Coulomb gauge](/source/Coulomb_gauge), Maxwell's equations are<ref>Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, {{ISBN|81-7758-293-3}}</ref>

:<math> \nabla^2 \varphi =-\dfrac{\rho}{\epsilon_0}</math>

:<math> \nabla^2 \mathbf{A} - \dfrac{1}{c^2}\dfrac{\partial^2 \mathbf{A}}{\partial t^2}=- \mu_0 \mathbf{J} +\dfrac{1}{c^2}\nabla\left(\dfrac{\partial \varphi}{\partial t}\right)\,,</math>

although the solutions contrast the above, since '''A''' is a retarded potential yet φ changes ''instantly'', given by:

:<math>\varphi(\mathbf{r}, t) = \dfrac{1}{4\pi\epsilon_0}\int \dfrac{\rho(\mathbf{r}',t)}{|\mathbf r - \mathbf r'|}\mathrm{d}^3\mathbf{r}'</math>

:<math> \mathbf{A}(\mathbf{r},t) = \dfrac{1}{4\pi \varepsilon_0} \nabla\times\int \mathrm{d}^3\mathbf{r'} \int_0^{|\mathbf{r}-\mathbf{r}'|/c} \mathrm{d}t_r \dfrac{ t_r \mathbf{J}(\mathbf{r'}, t-t_r)}{|\mathbf{r}-\mathbf{r}'|^3}\times (\mathbf{r}-\mathbf{r}') \,.</math>

This presents an advantage and a disadvantage of the Coulomb gauge - φ is easily calculable from the charge distribution ρ but '''A''' is not so easily calculable from the current distribution '''j'''. However, provided we require that the potentials vanish at infinity, they can be expressed neatly in terms of fields:

:<math>\varphi(\mathbf{r}, t) = \dfrac{1}{4\pi}\int \dfrac{\nabla \cdot \mathbf{E}(\mathbf{r}',t)}{|\mathbf r - \mathbf r'|}\mathrm{d}^3\mathbf{r}'</math>

:<math> \mathbf{A}(\mathbf{r},t) = \dfrac{1}{4\pi}\int \dfrac{\nabla \times \mathbf{B}(\mathbf{r}',t)}{|\mathbf r - \mathbf r'|}\mathrm{d}^3\mathbf{r}'</math>

==In linearized gravity==

The retarded potential in [linearized general relativity](/source/linearized_gravity) is closely analogous to the electromagnetic case. The trace-reversed tensor <math display="inline">\tilde h_{\mu\nu} = h_{\mu\nu} - \frac 1 2 \eta_{\mu\nu} h</math> plays the role of the four-vector potential, the [harmonic gauge](/source/harmonic_coordinate_condition) <math>\tilde h^{\mu\nu}{}_{,\mu} = 0</math> replaces the electromagnetic Lorenz gauge, the field equations are <math>\Box \tilde h_{\mu\nu} = -16\pi G T_{\mu\nu}</math>, and the retarded-wave solution is<ref>Sean M. Carroll, "Lecture Notes on General Relativity" ([https://arxiv.org/abs/gr-qc/9712019 arXiv:gr-qc/9712019]), equations 6.20, 6.21, 6.22, 6.74</ref>
<math display="block">\tilde h_{\mu\nu}(\mathbf r, t) = 4 G \int \frac{T_{\mu\nu}(\mathbf r', t_r)}{|\mathbf r - \mathbf r'|} \mathrm d^3 \mathbf r'.</math>
Using SI units, the expression must be divided by <math>c^4</math>, as can be confirmed by dimensional analysis.

==Occurrence and application==
A many-body theory which includes an average of retarded and ''advanced'' [Liénard–Wiechert potential](/source/Li%C3%A9nard%E2%80%93Wiechert_potential)s is the [Wheeler–Feynman absorber theory](/source/Wheeler%E2%80%93Feynman_absorber_theory) also known as the Wheeler–Feynman time-symmetric theory.

In [gravitation](/source/gravitation), there are application examples for calculating deviations in [orbits](/source/orbit) of [satellites](/source/Satellite),<ref>{{cite journal |author=C. K. Raju |title=Retarded gravitation theory |journal=AIP Conference Proceedings |date=2012 |volume=1483 |issue=1 |doi=10.1063/1.4756973 |pages=260–276 |arxiv=1102.2945 |bibcode=2012AIPC.1483..260R |url=https://pubs.aip.org/aip/acp/article-abstract/1483/1/260/727544/Retarded-gravitation-theory?redirectedFrom=fulltext |access-date=2024-11-25}}</ref> [moons](/source/Orbit_of_the_Moon)<ref>{{cite web |author=Yin Zhu |title=The speed of gravity: An observation on galaxy motions |date=2016 |doi=10.13140/RG.2.2.30917.45287 |url=http://rgdoi.net/10.13140/RG.2.2.30917.45287 |access-date=2024-11-25}}</ref> or [planets](/source/Orbits_of_planets).<ref>{{citation|access-date=2024-11-25 |author=Roy J. Kennedy |date=1929-09-15 |doi=10.1073/pnas.15.9.744 |issn=0027-8424 |issue=9 |pages=744–753 |periodical=Proceedings of the National Academy of Sciences |pmid=16577233 |title=PLANETARY MOTION IN A RETARDED NEWTONIAN POTENTIAL FIELD |url=https://pnas.org/doi/full/10.1073/pnas.15.9.744 |volume=15|pmc=522551 }}<!-- auto-translated from German by Module:CS1 translator --></ref> The anomalies in the [rotation curve](/source/rotation_curve)s of more than one hundred [spiral galaxies](/source/spiral_galaxies) of different [types](/source/Galaxy_morphological_classification) could also be explained. The data of the “SPARC (Spitzer Photometry and Accurate Rotation Curves) Galaxy collection”, which were recorded with the [Spitzer Space Telescope](/source/Spitzer_Space_Telescope), were used for this purpose. In this way, neither the assumption of [dark matter](/source/dark_matter) nor a modification of [general relativity](/source/general_relativity) is required to explain the observations.<ref>{{Cite journal |last1=Glass |first1=Yuval |last2=Zimmerman |first2=Tomer |last3=Yahalom |first3=Asher |date=2024-02-20 |title=Retarded Gravity in Disk Galaxies |url=https://www.preprints.org/manuscript/202402.1088/v1 |journal=Symmetry |language=en |volume=16 |issue=4 |pages=387 |doi=10.3390/sym16040387 |doi-access=free |bibcode=2024Symm...16..387G |issn=2073-8994 |access-date=2024-11-25}}</ref> On even larger scales, the retarded gravitational potentials result in effects such as an [accelerated expansion](/source/expansion_of_the_universe), which leads to an [isotropic](/source/isotropy), but not homogeneous [universe](/source/universe) with an outer shell of dark matter with an increased [mass density](/source/density) as well as a strong [gravitational redshift](/source/gravitational_redshift) of distant [astronomical objects](/source/astronomical_object).<ref>{{citation|access-date=2024-11-25 |author=Markus Bautsch |date=2024 |doi=10.13140/RG.2.2.27349.23529 |title=Retarded gravitational potentials on the scale of the universe |url=https://rgdoi.net/10.13140/RG.2.2.27349.23529}}<!-- auto-translated from German by Module:CS1 translator --></ref>

==Example==
The potential of charge with uniform speed on a straight line has [inversion in a point](/source/inversion_in_a_point) that is in the recent position. The potential is not changed in the direction of movement.<ref>[https://feynmanlectures.caltech.edu/II_26.html Feynman, Lecture 26, Lorentz Transformations of the Fields]</ref>

==See also==
* [Maxwell's equations](/source/Maxwell's_equations)
* [Liénard–Wiechert potential](/source/Li%C3%A9nard%E2%80%93Wiechert_potential)
* [Panofsky–Phillips equations](/source/Panofsky%E2%80%93Phillips_equations)
* [Jefimenko's equations](/source/Jefimenko_equations)
* [Lenz's law](/source/Lenz's_law)
* [Whitehead's theory of gravitation](/source/Whitehead's_theory_of_gravitation)

==References==

{{Reflist}}

Category:Potentials

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Adapted from the Wikipedia article [Retarded potential](https://en.wikipedia.org/wiki/Retarded_potential) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Retarded_potential?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.